This supplement to Document D (DOI: 10. 5281/zenodo. 19287163) provides three results concerning the topological displacement Δₜop = 3/20 for (5, 2) -Dehn surgery on the figure-eight knot 4₁ within Scalar-Tensor Knot-Web Cosmology (STKWC). (1) An independent derivation of Δₜop = 3/20 from the quadratic refinement of the surgery linking form, yielding the universal filling correction Δfill = q̄/ (4p) where q̄ = q⁻¹ mod p. This algebraic/arithmetic path is independent of the period-integral derivation in Document D. (2) A scope clarification for Document D's Theorem 3 ("surgical quadratic reciprocity"): the formula (q') ²·Π₊ (1) / (4π²N) is specific to (5, 2) -surgery on 4₁ and does not generalize to other slopes via the fixed full-oval period Π₊ (1) = 2π²/15. The universal formula for arbitrary primitive (p, q) is q̄/ (4p). (3) A reformulation of the filling correction within C0 (Contrast Calculus), the relations-first ontology underlying STKWC. Surgery is defined as contrast freezing — trivializing the holonomy along a chosen peripheral cycle. The filling correction emerges as residual cyclic curvature on an abstract rank-2 contrast lattice, requiring no 3-manifold. The agreement between the analytic/physical path (Document D) and the algebraic/arithmetic path (this supplement) constitutes independent verification that the (2+1) -dimensional Chern–Simons theory on the degeneracy surface ε = 0 produces invariants consistent with canonical surgery theory. Three open problems are formulated: a generalized surgery-dependent period formula, the non-abelian extension of C0 to SL (2, ℂ), and Two-Path verification for other hyperbolic knots. Computational verification performed within the Parliament of Dragons multi-AI methodology (Claude/Anthropic, ChatGPT/OpenAI, Gemini/Google DeepMind). Notes: Part of the STKWC preprint series. Document D Supplement v1. 0.
Yanush Feshter (Fri,) studied this question.